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http://functions.wolfram.com/06.33.21.0106.01
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Integrate[z^2 Log[b z] FresnelC[a z], z] ==
(1/(18 a^3 Pi^2)) ((-1 + I a^2 Pi z^2 + E^(I a^2 Pi z^2)
(-1 - I a^2 Pi z^2) + 3 E^((1/2) I a^2 Pi z^2)
(ExpIntegralEi[(-(1/2)) I a^2 Pi z^2] + ExpIntegralEi[
(1/2) I a^2 Pi z^2]) + 3 I (2 I - a^2 Pi z^2 +
E^(I a^2 Pi z^2) (2 I + a^2 Pi z^2)) Log[b z] +
2 a^3 E^((1/2) I a^2 Pi z^2) Pi^2 z^3 FresnelC[a z] (-1 + 3 Log[b z]))/
E^((1/2) I a^2 Pi z^2))
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 18 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> Ei </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mo> + </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <ci> FresnelC </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ln /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <imaginaryi /> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> ExpIntegralEi </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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